Let $f(t;x)$ and $g(x)=x$ for $x,f \in (0,1)$ and parameter $t \in \Bbb R_{\gt 0}.$ Is there a function that divides $R=(0,1)^2$ into pairs of equal area for all $t?$
Define $f(t;x)=(1-x)^t$ and $g(x)=x.$ Clearly for $t=1$ we are successful. But this is the only $t$ for which the constraints are met, so this choice fails.
Basically want $A+B=C$ for some constant $C$ that holds for all $t\in \Bbb R_{\gt 0}.$
